The most common convex approximation technique uses the \( \ell _1 \) -norm instead of the \( \ell _0 \) -norm in REF .
An overview on such \(\ell _1\) -surrogate models, their advantages ans solution approaches can be found in [1]}.
Provided that \( f \)  and \( X \)  themselves are convex, the resulting optimization problem is convex (though
nonsmooth) and can therefore be solved by a variety of methods for convex optimization, see [2]}.
This approach is very popular, for example, in solving compressed sensing problems.
On the other hand, there exist prominent applications, where the \( \ell _1 \) -norm provides absolutely no sparsity (like the portfolio optimization problem used in our numerical section).
